Don't try to use PgAdmin-III for this. Use pg_dump and pg_restore directly if possible.

Use the version of pg_dump from the destination server to dump the origin server. So if you're going from (say) 8.4 to 9.2, you'd use 9.2's pg_dump to create a dump. If you create a -Fc custom format dump (recommended) you can use pg_restore to apply it to the new database server. If you made a regular SQL dump you can apply it with psql.

You'll have a hard time creating an SQL dump that'll work in any version of PostgreSQL. Say you created a VIEW that uses a WITH query. That won't work when restored to PostgreSQL 8.3 because it didn't support WITH. There are tons of other examples. If you must support old PostgreSQL versions, do your development on the oldest version you still support and then export dumps of it for newer versions to load. You cannot sanely develop on a new version and export for old versions, it won't work well if at all.

More troubling, developing on an old version won't always give you code that works on the new version either. Occasionally new keywords are added where support for new specification features are introduced. Sometimes issues are fixed in ways that affect user code. For example, if you were to develop on the (ancient and unsupported) 8.2, you'd have lots of problems with implicit casts to text on 8.3 and above.

Your best bet is to test on all supported versions. Consider setting up automated testing using something like Jenkins CI. Yes, that's a pain, but it's the price for software that improves over time. If Pg maintained perfect backward and forward compatibility it'd never improve.